Dichotomy in temporal and thermal spin correlations observed in the breathing pyrochlore LiGa_1−xIn_xCr₄O₈

Abstract

A breathing pyrochlore system is predicted to host a variety of quantum spin liquids. Despite tremendous experimental and theoretical efforts, such sought-after states remain elusive as perturbation terms and lattice distortions lead to magnetic order. Here, we utilize bond alternation and disorder to tune a magnetic ground state in the Cr-based breathing pyrochlore LiGa_1−xIn_xCr₄O₈. By combining thermodynamic and magnetic resonance techniques, we provide experimental signatures of a spin-liquid-like state in x = 0.8, namely, a nearly T²-dependent magnetic specific heat and persistent spin dynamics by muon spin relaxation (μSR). Moreover, ⁷Li NMR, ZF-μSR, and ESR unveil the temporal and thermal dichotomy of spin correlations: a tetramer singlet on a slow time scale vs. a spin-liquid-like state on a fast time scale. Our results showcase that a bond disorder in the breathing pyrochlore offers a promising route to disclose exotic magnetic phases.

Introduction

A pyrochlore lattice provides a fertile ground to discover exotic quantum and topological phenomena since a three-dimensional (3D) network of corner-sharing tetrahedra features a macroscopic ground-state degeneracy^1,2,3,4,5,6. When quantum fluctuations are sufficiently strong, the pyrochlore Heisenberg antiferromagnet is predicted to harbor a spin-liquid ground state, whose defining characteristics are fractionalized excitations, emergent gauge fields, and long-range entanglement^7,8,9,10. However, such a quantum spin liquid (QSL) in pyrochlore materials has proven exceptionally difficult to achieve because a large coordination number suppresses quantum fluctuations and a range of perturbations such as spin-lattice couplings is apt to stabilize magnetic order^11,12,13.

In the context of a generalized pyrochlore lattice, a breathing pyrochlore has recently garnered much attention as a reservoir of exotic quantum states^14,15. In the breathing pyrochlore, A and B tetrahedra alternate in size and, thus, exchange interactions (see Fig. 1a). As such, its spin Hamiltonian is given by ${\cal{H}} = J_A\mathop {\sum }\nolimits_{i,j \in {\mathrm{A}}} {\mathbf{S}}_i \cdot {\mathbf{S}}_j + J_B\mathop {\sum }\nolimits_{i,j \in {\mathrm{B}}} {\mathbf{S}}_i \cdot {\mathbf{S}}_j$. Here, the breathing anisotropy B_f = J_B/J_A quantifies the degree of the bond alternation. By gauging inter-tetrahedral coupling J_B, decoupled tetrahedra (B_f = 0) is tuned to a regular pyrochlore (B_f = 1). At the endpoints, the ground states are tetramer singlet and QSL^9,16. The breathing pyrochlore has an advantage over its uniform counterpart in searching for QSL thanks to its strong thermal stability¹⁷. Besides, the inclusion of magnetic anisotropies to a pyrochlore Heisenberg model leads to a realization of Weyl magnons—a bosonic analog of a Weyl fermion in electron systems and a fracton spin liquid whose elementary excitations mimic electric and magnetic fields having the form of rank-2 tensors^18,19.

Fig. 1: Crystal structure, phase diagram, and magnetic properties.

a Schematic structure of the bond-disordered breathing pyrochlore LiGa_0.2In_0.8Cr₄O₈. b T-x phase diagram of LiGa_1−xIn_xCr₄O₈. The inverse red triangle marks the investigated x = 0.8. c Temperature dependence of the static magnetic susceptibility χ_dc(T) and its decomposition into the intrinsic χ_intr(T) and the orphan spin contribution χ_orp(T). d High-field magnetization at selected temperatures. The black dash-dotted and green dashed lines indicate the orphan spin magnetization M_orp(T) and the bulk magnetization M_intr(T), respectively. A linear extrapolation gives the spin pseudogap of Δ_M/k_B ~ 20 K.

On the material side, the Cr-based breathing pyrochlore LiACr₄O₈ (A = Ga³⁺, In³⁺; space group $F\bar 43m$) is a particularly interesting instance because B_f can be tuned by varying the composition of Li and A¹⁵. LiGaCr₄O₈, having the breathing anisotropy B_f = 0.6, behaves much like a uniform pyrochlore. On the other hand, LiInCr₄O₈ has the small B_f = 0.1, in which the A tetrahedra are weakly coupled through the inter-tetrahedral interaction J_B. Despite the large difference in B_f, both the LiACr₄O₈ compounds show commonly two-stage magneto-structural transitions at T_S = 16 (17) K and T_M = 13 (14) K for A = In (Ga)²⁰. The lasting antiferromagnetic orders suggest that the bond alternation of the x = 0 and x = 1 compounds is not sufficient to repress the effect of spin-lattice coupling that is responsible for a cubic-to-tetragonal transition accompanying structural and magnetic phase segregation. In this situation, bond randomness can counteract the magnetoelastic transition by inhibiting cooperative lattice distortions. This raises the exciting prospect of unveiling concealed magnetic behaviors of the breathing pyrochlore lattice, not being interrupted by long-range magnetic order.

In this work, we investigate spin dynamics and low-energy excitations of the bond-disordered breathing pyrochlore LiGa_0.2In_0.8Cr₄O₈ (marked by the inverse triangle in Fig. 1b). By combining multiple magnetic resonances with thermodynamic techniques, we find a thermal and temporal dichotomy between a coupled tetramer singlet and a spin-liquid-like state with no indication of the substitution-induced phase coexistence. Spin fluctuations probed on the different time scales evidence that the QSL-like state is pertinent to an emergent diluted pyrochlore lattice by a random bond alternation.

Results

Magnetic phase diagram

As sketched in Fig. 1a, the A-site ordered LiACr₄O₈ (A = Ga, In) spinels are a nearly perfect realization of the breathing pyrochlore lattice due to the significant difference between the Li⁺ and A³⁺ valence states. The Ga-for-In substitution disrupts a uniform arrangement of the two alternating tetrahedra and leads to the rich magnetic phase diagram of LiGa_1−xIn_xCr₄O₈, as shown in Fig. 1b. The magnetic ordering vanishes for x > 0.1 or x < 0.95. A spin-glass phase occupies a wide range of x = 0.1–0.625. For the Ga-rich compound LiGa_0.95In_0.05Cr₄O₈, the ⁷Li NMR and neutron experiments revealed the spin nematic transition at T_f = 11 K, driven by the small bond disorder. In the In-rich range of x = 0.75–0.95, LiGa_1−xIn_xCr₄O₈ exhibits a spin-gap-like behavior with the lack of magnetic ordering and spin freezing down to 2 K. However, little is known about an exact ground state and spin dynamics. Here, we have chosen the x = 0.8 compounds (marked by the inverse triangle in Fig. 1b) to address this issue.

Magnetic susceptibilities and magnetization

Figure 1c presents the T dependence of dc magnetic susceptibility χ_dc(T) of x = 0.8. On cooling, χ_dc(T) shows a broad hump at around 70 K, and a subsequent upturn below 10 K. Above 100 K, χ_dc(T) follows the Curie-Weiss law with the Curie-Weiss temperature Θ_CW = −386(1) K and the effective magnetic moment μ_eff = 3.873(4) μ_B [see Supplementary Note 1] that are consistent with the previously reported values²¹. As the first step to single out the intrinsic χ_intr(T), we attempted to fit the T < 5 K data to χ_dc(T) = χ_intr(T)+ χ_orp(T), where χ_orp(T) = C_orp/(T–Θ_orp). We obtain Θ = −2.0(2) K, alluding to the contribution of weakly interacting orphan spins. The orphan spin concentration is estimated to be 0.6% of the Cr³⁺ spins. Defects and chemical disorders induced by the chemical substitution may be responsible for the appearance of a tiny fraction of the weakly interacting orphan spins. The resulting χ_intr(T < 5 K), which is nearly T-independent, indicates the presence of abundant low-energy in-gap states. Besides, the ac magnetic susceptibility χ‘_ac(T) excludes the formation of spin freezing or spin glass state [see Supplementary Note 2].

Figure 1d displays the high-field magnetization M(T, B) measured up to 58 T at selected temperatures. With increasing field, M(T = 1.8 K, B) shows an upward convex behavior at low fields and then switches to a concave curvature at high fields. The low-B convex M(B) is described by a Brillouin function B_J, yielding 1.5% of the orphan spins. By subtracting M_orp(B) from M(B), we obtain the intrinsic magnetization M_intr(B), as illustrated by the green line in Fig. 1d. The linear extrapolation of the linearly increasing M_intr(B) above 30 T yields the spin pseudogap Δ_M/k_B~ 20 K, comparable to the previous result²². The finite slope of M_intr(B) at low fields is associated with the in-gap state, consistent with the χ_dc(T) and χ′_ac(T) [see Supplementary Note 2]. At T = 60 K (≫Δ_M/k_B), M(B) shows a quasilinear increase, characteristic of antiferromagnetically coupled spins.

Heat capacity

In Fig. 2, we plot the heat capacity C_tot of x = 0.8 as a function of T and B. We subtract lattice and nuclear Schottky contributions from C_tot to isolate the magnetic specific heat C_m [see Supplementary Note 3 for details]. As evident from Fig. 2a, C_m shows a broad maximum around 70 K, indicative of the onset of short-range ordering, and is largely quenched above 150 K.

Fig. 2: Heat capacity of the breathing pyrochlore LiGa_0.2In_0.8Cr₄O₈.

a Temperature dependence of the total heat capacity C_tot in various magnetic fields B = 0, 1, 3, 6, and 9 T. The blue dashed curve denotes the evaluated lattice contributions using the Debye model. The open pink circles represent the magnetic heat capacity C_m. b T and B dependence of the magnetic heat capacity divided by temperature C_m/T on a double log scale. The dashed lines are fit to the power-law behavior C_m/T ~ T ⁿ with n = 0.68–0.82. The inset shows a comparison between the C_m/T data and the theoretical curve for a s = 3/2 tetrahedron model.

From a log-log plot of C_m/T vs. T in Fig. 2b, we observe a hump at around T = 0.6 K in addition to the high-T broad hump. On the application of an external magnetic field, the T = 0.6 K hump shifts to a higher temperature and is quickly suppressed at μ₀H > 3 T. The B-dependence of the low-T hump is attributed to the weakly coupled orphan spins, which occupy the lowest m_s level by thermal population. Note that the spin entropy S_m has a value of 0.186 J·K⁻¹·mol⁻¹·Cr⁻¹ at 0.6 K, which amounts to roughly 0.2% of Rln4 [see Supplementary Note 3 for details]. This value is somewhat smaller than the estimated value from χ_dc(T) and M(T = 1.8 K, B).

In addition, C_m(T)/T displays a moderate B dependence in the T range of T = 0.1–10 K. Overall, the C_m(T)/T data are described by the power-law dependence C_m/T ~ Tⁿ with n = 0.68–0.82 at T = 2–10 K and n = 0.67–0.86 below 1 K. The observed C_m~ T^1.68–1.82 is close to a quadratic dependence. We recall that the T² dependence of C_m has been reported in a range of 3D frustrated magnets, which are commonly proximate to a disordered QSL with unconventional low-energy excitations^23,24,25. A comparison between the C_m/T data and the calculated curve of a s = 3/2 isolated tetrahedron unveils a large discrepancy for temperatures below 10 K [see the inset of Fig. 2b]. Furthermore, the spin gap is evaluated to be Δ/k_B = 0.289(5) K from the low-T heat capacity data, suggesting that the excitation gap by tetramer singlets is filled by the in-gap magnetic states at low temperatures [see Supplementary Note 3 for details]. Thus, we conclude that the observed quadratic dependence of C_m is associated with emergent gapless low-energy excitations of the bond-disordered breathing pyrochlore and that the orphan spin contribution is minor in forming a spin-liquid-like state, judging from the fact that the gapless excitations develop at temperature as high as 10 K, well above T = 0.6 K.

Nuclear magnetic resonance

We now examine temporal and thermal spin fluctuations by combining multiple magnetic resonance techniques.

The ⁷Li NMR spectra and the spin-lattice (spin–spin) relaxation rate 1/T₁ (1/T₂) measured at μ₀H = 11.99 T are plotted as a function of temperature in Fig. 3a, e. At high temperatures, the NMR spectrum exhibits a single sharp line with no quadrupolar splitting, similarly observed in LiInCr₄O₈^20,26. On cooling down to T = 2.7 K, the NMR spectra gradually broaden and shift to higher magnetic fields without developing additional structured peaks, further confirming the absence of long-range magnetic ordering and structural phase transition.

Fig. 3: Thermal and temporal evolution of spin fluctuations of LiGa_0.2In_0.8Cr₄O₈ measured by ⁷Li NMR, ZF-μSR, and ESR.

a ⁷Li NMR spectra at selected temperatures T = 2.7 and 250 K. b ZF-μSR spectra at selected temperatures. c Longitudinal-field dependence of the μSR spectra measured at T = 25 mK. The solid lines denote the fit to the data using the sum of two simple exponential functions. d Derivative of the ESR absorption spectra measured at various temperatures. Spectra are vertically shifted for clarity. e T dependence of ⁷Li spin-lattice relaxation rate 1/T₁ and spin-spin relaxation rate 1/T₂. The red solid line denotes the Arrhenius equation, yielding the spin pseudogap of Δ/k_B = 35(1) K. The solid black line represents a power-law behavior of 1/T₁ and 1/T₂. f Muon spin relaxation rate as a function of temperature λ_f(T). The solid magenta line represents the power-law fit to the data. g Muon spin relaxation as a function of longitudinal field λ_LF(H), along with fitting to the Redfield formula. h T dependence of the peak-to-peak ESR linewidth ΔH_pp(T). The solid orange lines indicate a power-law behavior of ΔH_pp(T) in two different temperature ranges marked by the shaded region at T^* = 10 K. Error bars on the data points indicate a standard error.

With decreasing temperature, 1/T₁ displays an exponential-like decrease down to 10 K, and then a subsequent upturn to a power-law growth. The activation behavior of 1/T₁ in the T range of 25–100 K is well fitted with the Arrhenius equation 1/T₁~ exp(−Δ/k_BT) [see the semi-log plot of 1/T₁ vs. 1/T in the inset of Fig. 3e]. We obtain the spin pseudogap Δ/k_B = 35(1) K, somewhat larger than Δ_M/k_B~20 K extracted from M_intr(B). Generally, the spin gap Δ/k_B obtained from 1/T₁ is larger than the true spin gap Δ_min(q) occurring at a specific point q^27,28,29. This is because 1/T₁ maps out the q-average of the dynamical spin susceptibility, 1/T₁ ~ T Σ_qA²(q)χ“(q,ω₀). Here, A(q) is the form factor of hyperfine interactions, and ω₀ is the nuclear Larmor frequency.

The salient feature is that 1/T₁ follows a power law 1/T₁ ~ T⁻ⁿ with the exponent of n = 0.46(3) below T^* = 10 K. The drastically distinct behavior of 1/T₁ through T^* = 10 K suggests the switching of a dominant relaxation mechanism: a high-T activated vs. a low-T fast fluctuating. Here the small exponent n is indicative of dynamically fluctuating spin excitations, defying the development of a λ-like critical slowing down. The spin-spin relaxation rate 1/T₂ shows a weak power-law decrease 1/T₂ ~ T ^0.071(4) as T → 0. A nearly T-independent 1/T₂ means that spin dynamics is in the fast-fluctuating limit without critical slowing down, as observed in the QSL states of the triangular material 1T-TaS₂³⁰.

Muon spin relaxation (μSR)

In Fig. 3b, we present ZF-μSR spectra at selected temperatures. At high temperatures, the ZF-μSR spectra show a Gaussian-like shape and gradually change to an exponential-like form. With decreasing temperature, the muon spin polarization rapidly relaxes in the initial time interval (t = 0–2 μs). We observe neither an oscillating muon signal down to T = 25 mK nor a recovery to 1/3 of the initial polarization, evidencing the formation of a dynamically fluctuating state³¹. No loss of the initial polarization is not compatible with inhomogeneous magnetism.

For quantitative analysis, we fitted the ZF-μSR spectra to the sum of a stretched exponential function and an exponential function, P_z(t) = a_fastexp[−(λ_ft)^β] + a_slowexp[−λ_st]. Here, a_fast (a_slow) is the fraction of the fast (slow) relaxation component, and λ_f (λ_s) is the muon spin relaxation rate for the fast (slow) relaxing component. We summarize the obtained parameters in Fig. 3f and Supplementary Note 4. As the temperature decreases, the slow relaxation component λ_s(T) suddenly appears below T^* = 10 K, at which ⁷Li 1/T₁ shows an upturn to the power law. Furthermore, λ_s(T) shows a T-independent behavior (λ_s ~ 0.042 μs⁻¹) below 1 K.

Figure 3f is a log-log plot of the fast relaxation rate vs. T. With lowering the temperature, λ_f(T) exhibits a weak power-law increase T^−0.11(2) down to 10 K, and then a steep power-law increase with the exponent of n = −0.71(6). Noteworthy is that μSR probes weakly correlated spins in the T > 10 K range where NMR senses gapped spin excitations, implying the variation of spin dynamics with a detecting time scale. We further note that the steep increment of λ_f(T) concurs with the leveling-off of χ_intr(T) in Fig. 1c and the upturn of 1/T₁ in Fig. 3e. Upon further cooling toward T = 25 mK, λ_f(T) flattens out below 2 K, entering a persistent-spin-dynamical regime. Such a leveling-off of the muon spin relaxation rate is often observed in an assortment of QSL candidates^32,33,34. With decreasing temperature, the stretched exponent β gradually decreases from β = 2 (a Gaussian-like decay) to β = 1 (an exponential decay) below T^* = 10 K [see Supplementary Note 4, Fig. S4]. The low-T exponential relaxation suggests the persisting spin fluctuations to T = 25 mK without forming frozen moments.

Shown in Fig. 3c and g are the LF-μSR spectra and the LF dependence of λ_f(H_LF) measured at T = 25 mK. Remarkably, the LF-μSR spectra exhibit substantial relaxation even at H_LF = 10 kG, lending further support to the dynamic ground state. In the Redfield model, ${\lambda}_{\mathrm{LF}}{({H})} = 2{\gamma}_{\mu}^{2} \langle {H}_{\mathrm{loc}}^{2} \rangle {\nu}/ ({\nu}^{2} + {\gamma}_{\mu}^{2} {H}^{2})$, the LF dependence of λ_f(H_LF) is related to the fluctuation frequency ν, and the fluctuating time-averaged local field $H_{{\mathrm{loc}}}^2$³⁵. Here, γ_μ is the muon gyromagnetic ratio. Fitting to the λ_f(H_LF) data yields ν = 1.6(1) GHz and H_loc = 296(11) G [see Fig. 3g]. We comment that the fluctuation frequency is faster than the value reported thus far for QSL materials while the magnitude of the local field is rather large, possibly due to a large spin number s = 3/2^32,33.

Electron spin resonance

To probe spin fluctuations on the GHz time scale, we turn to high-frequency ESR. As shown in Fig. 3d, with decreasing temperature, the ESR spectra broaden continuously down to 2.3 K and shift to a lower field. We observe no impurity signals, excluding the possibility of the Ga³⁺/In³⁺ substitution-induced phase separation. All the ESR spectra are fitted with a single Lorentzian profile, suggesting that they originate from one type of the correlated Cr spins. The extracted parameters vs. temperature are plotted in Fig. 3h and Supplementary Note 5, Fig. S5. The peak-to-peak linewidth ΔH_pp is associated with the development of spin-spin correlations, while the resonance field H_res emulates the build-up of internal magnetic fields.

ΔH_pp(T) exhibits a power-law increase ΔH_pp(T) ~ T^-n. We can identify a small change of the exponent from n = 0.37(1) to 0.30(2) through T^* = 10 K. This weak anomaly in the exponent is in sharp contrast to the drastically changing character of the magnetic correlations on the MHz time scale. A critical-like line broadening in the paramagnetic state is due to the development of local spin correlations below Θ_CW = −386(1) K and is generic to frustrated magnets subject to critical spin fluctuations. We note that the observed critical exponent of n = 0.30–0.37 is much smaller than n = 0.6–0.7 for the uniform pyrochlore MCr₂O₄ (for M = Zn, Mg, Cd)³⁶, as shown in Fig. 3h. Compared to the uniform pyrochlore, the reduced exponent in x = 0.8 means that the random bond alternation weakens critical spin correlations. As such, the emergent low-T state in the GHz time scale bears a resemblance to a diluted pyrochlore lattice.

Raman spectroscopy

Next, we discuss the structural evolution of LiGa_1−xIn_xCr₄O₈ with Raman spectroscopy. We present the composition dependence of the Raman spectra and phonon parameters in Fig. 4. Herein, we focus the T_2g(1) and T'_2g(1) phonon modes, which are sensitive to the structural environments of the A and B tetrahedra, respectively [see Supplementary Note 7 and Fig. S7 for the phonon assignments].

Fig. 4: Comparison of the room-temperature low-energy Raman spectra for LiGa_1−xIn_xCr₄O₈.

a Composition dependence of the T_2g(1) and T'_2g(1) phonon modes of LiGa_1−xIn_xCr₄O₈ measured at room temperature. The dashed lines are guides to the eye. b Phonon parameters of the T_2g(1) and T'_2g(1) modes as a function of x.

As shown in Fig. 4a, b, the 314 cm⁻¹ T'_2g(1) mode slightly softens with increasing x to 0.5 and then little changes with further In³⁺ substitution. Its linewidth decreases linearly with x despite the introduction of Ga³⁺/In³⁺ chemical disorders. This suggests that a single type of A tetrahedra (surrounded by four Li⁺ ions) is present for all x. In sharp contrast, the T_2g(1) mode at 216 cm⁻¹ is split into three peaks for x = 0.5 and 0.8. For a random occupation of the In³⁺ and Ga³⁺ ions, five types of the B tetrahedra surrounded by a combination of the In³⁺ and Ga³⁺ ions are possible in LiGa_0.2In_0.8Cr₄O₈: four In³⁺ (4I), four Ga³⁺ (4G), three In³⁺/one Ga³⁺ (3I1G), 2G2I, or 1G3I. For x = 0.8, the three 118, 160, and 195 cm⁻¹ modes have their scattering intensity in a relative ratio of 1:1.3:0.7. Given a low probability of occupying the 3G1I and the 4 G configurations, the three peaks are assigned to 4I, 3I1G, and 2I2G, respectively. From the composition-dependent Raman data, we infer that the In³⁺/Ga³⁺ substitution engenders a single type of the A tetrahedra and three types of the B tetrahedra. However, we admit that the chemical substitution can give rise to diverse effects in addition to the primary chemical pressure, including crystal-electric-field distortion and inhomogeneous charge distribution near the Cr³⁺ ions. Based solely on the Raman data, it is difficult to rationalize their impact on structure and magnetism. Future high-resolution x-ray studies are required to address this issue.

In Fig. 5a, we zoom the x evolution of the T_2g(3) phonon mode, which involves the asymmetric bending of the CrO₆ octahedra. As such, the T_2g(3) mode is sensitive to the substitution-induced alteration of the Cr-O bonding geometry, allowing tracing the randomness in exchange interactions. As plotted in Fig. 5b, the phonon frequencies of the T_2g(2), T_2g(3), and A_1g modes show a linear decrease with increasing x. A close look at the T_2g(2) and T_2g(3) modes reveals the splitting of the T_2g(2) and T_2g(3) modes on the In³⁺/Ga³⁺ substitution [see the decomposition of the phonon peaks into four Lorentzian profiles in Fig. 5a and the pink triangles in Fig. 5b]. The linewidths of the main peaks show little variation with x. This suggests that In³⁺/Ga³⁺ substitution does not bring about strong disorders such as vacancies, defects, and interstitial cations. Rather, the distinct distortion field induced by the In³⁺/Ga³⁺ substitution leads to the differentiation of Cr–O bonding environments. From the observation that most of the high-frequency phonons are split into two peaks, we conclude a differentiation of the intra-tetrahedral J_A, weak inter-tetrahedral J_B1, and strong inter-tetrahedral exchange interaction J_B2. That is, the bond randomness creates at least exchange interactions J_A, J’_A, J_B2, J’_B2, J_B1, and J’_B1 with an energy hierarchy J_A, J’_A > J_B2, J’_B2 > J_B1, J’_B1. Assuming the variations of δJ_A = J_A− J’_A, δJ_B1, and δJ_B2 are not bigger than the J_A and J_B2, the energy hierarchy is largely preserved.

Fig. 5: Composition evolution of high-energy phonons for LiGa_1-xIn_xCr₄O₈.

a Magnified view of the T_2g(3) phonon mode as a function of x. b Composition dependence of the phonon parameters for the T_2g(2), T_2g(3), and A_1g modes.

Discussion

With the aid of diverse thermodynamic and magnetic resonance techniques (0 Hz–1 THz), we are able to disclose the temporal and thermal structures of magnetic correlations.

We identify the characteristic temperature T^* = 10 K, below which highly correlated spins develop a quantum paramagnetic state. The emergent low-energy state features nearly T-squared C_m, as well as correlated spin fluctuations over a wide time scale of 10⁻¹²− 10⁻⁴ s, as evident from a power-law law 1/T₁ ~ T^−0.46(3), λ_f(T) ~ T^−0.71(6), and ΔH_pp(T) ~ T^−0.30. The exponents extracted from different resonance techniques cannot be related to each other since the breathing pyrochlore model is not analytically solvable. Nonetheless, the common power-law behaviors advocate critical-like spin fluctuations. Here we stress that the energy scale of T^* = 10 K is smaller than the spin pseudogap of Δ/k_B = 20–35 K, deduced from M(B) and 1/T₁. Yet, it is bigger than a few kelvins of weakly correlated orphan spins. The latter contribution is quenched by applying the external field of µ₀H > 3 T. As such, the contribution of defect spins to the low-T correlated state is minute.

Next turn to the temporal spin dynamics. The thermodynamic data show the spin-gap behavior with the in-gap state. On the slow time scale (1–10⁴ Hz), the NMR data demonstrate dichotomic spin dynamics, namely, high-T gapped spin correlations and low-T fast fluctuations. On the fast time scale (10⁴–10¹² Hz), the µSR and ESR data give no signature of the spin-singlet correlations. Instead, highly correlated spin dynamics prevails over the whole measured temperature range. It seems that the singlet fluctuations average out beyond the time scale of the NMR technique at low temperatures.

The dichotomic magnetic correlations are apt to be correlated with structural and magnetic inhomogeneities. First, we mention that the spectral shape and the relaxation rates of the magnetic resonance data rule out the coexistence of magnetically phase-segregated regions. Unlike the end members (x = 0, 1), 1/T₁(T), λ_f(T), and ΔH_pp(T) of x = 0.8 show no notable anomalies such as critical divergence, abrupt drop, or kink^20,26.

Second, the Raman spectra exhibit no phonon anomalies pertaining to magnetostructural transitions [see Supplementary Note 7 for details]. Instead, as sketched in Fig. 6, a statistical distribution of In³⁺ and Ga³⁺ generates mainly a single type of A tetrahedra and three types of tetrahedra: four-neighbor A tetrahedra are connected by (i) four weak inter-tetrahedral couplings J_B1s, (ii) three weak J_B1s and one strong J_B2, and (iii) two weak J_B1s and two strong J_B2s. Given the magnetic energy hierarchy J_A > J_B2 > J_B1 with J_B1/J_A = 0.3 and J_B2/J_A = 0.6, two magnetic subsystems emerge, namely (i) coupled tetramer singlets (red shaded circle) and (ii) diluted pyrochlore lattices (blue dashed region). Here, we note that J_B1/J_A increases from 0.1 to 0.3 in LiInCr₄O₈ through the magnetostructural transition with decreasing temperature³⁷. The tetramer singlets forming an entangled state of four spins are coupled through J_B1 and tetrahedral clusters. This bond randomness leads to a broadening and distribution of the discrete energy levels, yet still maintaining a singlet character. Taken together, the coupled tetramer singlets have longer temporal magnetic correlations than the pyrochlore-like subsystem that are governed by two-spin correlations. This raises the possibility that the quantum paramagnetic ground state bears a QSL-like characteristic pertinent in the diluted pyrochlore system. Here, we stress that despite the In³⁺/Ga³⁺ substitution-induced bond disorders, our thermodynamic and resonance data are not compatible with a spin glass, spin freezing, and random singlets (valence bonds). Of particular, we could find no scalings of thermodynamic quantities expected for random singlets^38,39,40. Rather, LiGa_0.2In_0.8Cr₄O₈ features a quantum disordered state with gapless low-lying excitations. We, thus, speculate that a degree of randomness is insufficient to engender random singlets or spin glass due to the presence of the tetramer-singlet-like state.

Fig. 6: Illustration of the emergent magnetic lattices in a bond-disordered breathing pyrochlore.

Schematics of single-A tetrahedra (orange squares) and three different B tetrahedra (green squares) created by the Ga-for-In substitution. The thick and thin green lines denote J_B2 and J_B1, respectively. Given the energy hierarchy of magnetic interactions J_A > J_B2 > J_B1, the breathing pyrochlore effectively comprises coupled tetramer singlets (red shaded circle) and diluted pyrochlore lattices (blue dashed region).

To conclude, we have investigated the temporal and thermal spin dynamics of a bond-disordered breathing pyrochlore compound. We find that a dichotomy of magnetic correlations is linked to two emergent subsystems. LiGa_0.2In_0.8Cr₄O₈ occupies a special position in QSL candidate materials because quenched disorders do not lead to a spin glass and random singlet. The observed low-T QSL-like state opens up a venue for discovering an unknown state in a diluted breathing pyrochlore lattice.

Methods

Sample synthesis

Polycrystalline samples of LiGa_0.2In_0.8Cr₄O₈ were synthesized by a solid-state reaction method. Stoichiometric amounts of Li₂CO₃, Cr₂O₃, Ga₂O₃, and In₂O₃ were mixed in a 1:4:0.2:0.8 molar ratio and thoroughly ground in a mortar. The mixture was pelletized and sintered at 800 °C in the air for 12 h. The substance was ground, pressured into pellets, and finally sintered at 1000 °C for 24 h and 1100 °C for 72 h. We checked the quality of the samples by X-ray diffraction measurements.

Magnetic properties characterization

dc and ac magnetic susceptibilities were measured using a superconducting quantum interference device magnetometer and vibrating sample magnetometer (Quantum Design MPMS and VSM). High-field magnetization experiments were performed at the Dresden High Magnetic Field Laboratory using a pulsed-field magnet (20 ms duration). The magnetic moment was detected by a standard inductive method with a pick-up coil system in the field range of μ₀H = 0–60 T.

Heat capacity

Heat capacity measurements were carried out in the temperature and field range of T = 0.03–300 K and B = 0–9 T using a commercial set-up of Quantum Design PPMS with a thermal relaxation method. The raw heat capacity data C_tot/T follow a T⁻³ power-law behavior at extremely low temperatures, corresponding to a nuclear Schottky contribution (see Supplementary Note 3). After subtracting the nuclear Schottky contribution C_NS/T = AT⁻³ from C_tot/T, we present the magnetic and lattice contributions to the specific heat in the main text.

Nuclear magnetic resonance

⁷Li (I = 3/2, γ_N = 16.5471 MHz/T) NMR measurements were conducted at National High Magnetic Field Laboratory (Tallahassee, USA) by using a locally developed NMR spectrometer equipped with a high-homogeneity 17 T sweepable magnet. ⁷Li NMR spectra were recorded by a fast Fourier transform of spin-echo signals while sweeping the field at a fixed frequency ν = 198.317 MHz. The nuclear spin-lattice (spin-spin) relaxation time T₁ (T₂) was measured by a modified inversion recovery (Hahn pulse) method width π/2 = 1 μs in the temperature range of T = 2.6–250 K.

Muon spin relaxation (μSR)

μSR measurements were carried out on the DR spectrometer (³He/⁴He dilution refrigerator, 25 mK ≤ T ≤ 10 K) at M15 beamline and the LAMPF spectrometer (⁴He cryostat, 2 < T < 300 K) at M20 beamline in TRIUMF (Vancouver, Canada).

The polycrystalline samples of LiGa_0.2In_0.8Cr₄O₈ were wrapped with a silver foil and then attached to the sample holder. After the mounted samples were inserted into the cryostat, μSR spectra were measured in zero-field, longitudinal field (parallel to muon spin direction), and transverse field (perpendicular to muon spin direction).

Electron spin resonance (ESR)

High-frequency ESR experiments were performed at National High Magnetic Field Laboratory (Tallahassee, USA). The ESR spectra were recorded using a quasioptical heterodyne spectrometer in the temperature range of T = 2–290 K⁴¹, enabling the detection of a magnetic field derivative of a microwave absorption signal. An external magnetic field was swept with a 12.5 T sweepable superconducting magnet at the fixed frequency ν = 240 GHz.

Raman spectroscopy

The Raman spectra were obtained with a micro-Raman spectrometer (XperRam 200, Nanobase Inc.). Samples were excited in exact backscattering geometry by using a 532 nm laser.